Schulze Method - Simpler Definition

Norman Petry npetry at sk.sympatico.ca
Tue Sep 8 13:09:10 PDT 1998

```Markus has suggested a couple of small clarifications to my definition of
his method which I thought I'd pass along:

>I want to propose two changes in the definition of the
>Schulze method.
>
>*****
>
>First: It should be clarified, that whenever a ballot is
>chosen randomly it is chosen among those ballots that
>haven't yet been chosen randomly.
>
>The reason: Someone could argue, that it could happen, that
>always the same ballots are chosen randomly, so that the
>Schulze method doesn't lead to a result in a finite amount
>of time.
>
>*****
>
>Second: It should be clarified, that whenever every ballot
>has been chosen randomly and there is still more than one
>winner, then the winner is chosen randomly among the
>potential winners.
>
>*****

One way to incorporate these changes in a description of the _complete_
Schulze method is to use the notation that was developed on this list some
time ago for compound methods.  For example:

Smith//Condorcet

meant the Smith method followed by the "Condorcet" method, where Condorcet's
method in this case was intended to mean the candidate with the fewest
votes-against in his/her worst pairwise loss (this is sometimes called the
"Minimax" rule).  It's understood using this notation that when one stage in
the compound method results in a tie, the tied subset of potential winners
are run through the overall method again, until a single-winner is found.

Using this notation, the complete Schulze method would be:

Schulze's Method = Beat-Path Undefeated//Random Ballot//Random.

with "Beat-Path Undefeated" as the core of the method.  Alternately, if
there are a large number of candidates, the Smith set could be used as the
initial filter, since Smith is easier to compute than Beat-Path Unbeaten.
In that case, the method is:

Schulze's Method = Smith//Beat-Path Undefeated//Random Ballot//Random.

In any case, the two definitions are equivalent, since Beat-Path Undefeated
will never select winners outside the Smith set anyway.

***

Finally, some (obvious) definitions:

Random Ballot:

Select a random ballot and remove it from the election.  The winners are the
candidates ranked highest on this ballot who have not yet been eliminated.

Random:

Select the winner randomly from candidates not yet eliminated.

***

Norm Petry

```